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Unformatted text preview: CSE 460  Solutions to Sample Final Exam Fall Semester, 2007 Name The type A points on this exam sum to 13. The type B points on this exam sum to 26. The type C points on this exam sum to 17. The type D points on this exam sum to 22. There are 3 extra credit points that can be applied to any category. 1. Consider the following grammar G 1 defined in abbreviated format. E → E + E  E * E  4  5  6  7 Given the following leftmost derivation of the string 4+5 * 6+7, answer the following questions. E ⇒ E + E ⇒ E * E + E ⇒ E + E * E + E ⇒ 4+ E * E + E ⇒ 4+5 * E + E ⇒ 4+5 * 6+ E ⇒ 4 + 5 * 6 + 7. (a) Draw the corresponding parse tree. [2, A] E /\ /  \ E + E /\  /  \  E * E 7 /\  /  \  E + E 6     4 5 (b) Draw the corresponding rightmost derivation. [2, A] E ⇒ E + E ⇒ E +7 ⇒ E * E +7 ⇒ E * 6+7 ⇒ E + E * 6+7 ⇒ E +5 * 6+7 ⇒ 4+5 * 6+7. (c) Give the numerical value of 4 + 5 * 6 + 7 implied by the above parse tree. [1,A] 9 * 6 + 7 = 61 (d) Give an alternative leftmost derivation for 4 + 5 * 6 + 7. [2,A] E ⇒ E + E ⇒ 4 + E ⇒ 4 + E * E ⇒ 4 + 5 * E ⇒ 4 + 5 * E + E ⇒ 4 + 5 * 6 + E ⇒ 4 + 5 * 6 + 7. (e) Correct the following incorrect definition of an ambiguous CFG G . [2,A] A CFG G is ambiguous if G has two or more distinct derivations for some string 1 x ∈ L ( G ). A CFG G is ambiguous if G has two or more distinct leftmost or rightmost derivations for some string x ∈ L ( G ). 2. Given the following two grammars G 1 and G 2 , apply the constructions we have seen in class to generate a new grammar G 3 such that L ( G 3 ) = ( L ( G 1 ) L ( G 2 )) * . Specify G 3 completely (not in abbreviated format). [4, B] G 1 G 2 V 1 = { S,T } V 2 = { S,T } Σ = { a,b } Σ = { a,b } Start variable S 1 is S Start variable S 2 is T P 1 = S → aSa  aT, T → bbST  b P 2 = S → bT  aS, T → ST  abab Grammar G 3 • V 3 = ( S 1 ,T 1 ,S,T,U,V ) • Σ = { a,b } • Start variable S 3 is V • P 3 = V → UV  λ U → S 1 T S 1 → aS 1 a  aT 1 T 1 → bbS 1 T 1  b S → bT  aS T → ST  abab 3. Suppose we apply the construction covered in class to construct an FSA M 2 from an NFA M 1 with 10 total states and 3 accepting states such that L ( M 2 ) = L ( M 1 ). (a) How many total states does M 2 have? [1, B] 2 10 (b) How many accepting states does M 2 have? [1, B] 2 9 + 2 8 + 2 7 2 4. Chomsky Normal Form (a) Identify all the nullable variables in the following grammar G 1 with start variable S . [1,B] S → ABCD  a A → BCD  BC  b B → CC C → a  b  λ D → aa  DD Nullable variables: A , B , and C (b) Apply the algorithms from class to produce a grammar G 2 such that L ( G 2 ) = L ( G 1 ) where G 2 has no λproductions. [1,B] S → ABCD  ABD  ACD  BCD  AD  BD  CD  D  a A → BCD  BD  CD  D  BC  B  C  b B → CC  C C → a  b D → aa  DD (c) Apply the algorithms from class to produce a grammar G 3 by eliminating the unit productions from the grammar G 2 . [1,B] S → ABCD  ABD  ACD...
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This note was uploaded on 07/25/2008 for the course CSE 460 taught by Professor Torng during the Fall '07 term at Michigan State University.
 Fall '07
 TORNG

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